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\define\e{\eta}
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\topmatter
\title
Agreement percolation and phase coexistence \\
in some Gibbs systems
\endtitle
\leftheadtext\nofrills
{G. Giacomin, J.L. Lebowitz, C. Maes}
\rightheadtext\nofrills
{Percolation in Gibbs states}
%
%
\author
G. Giacomin, J.L. Lebowitz and C. Maes
\endauthor
\affil
Department of Mathematics, Rutgers University and
Instituut voor Theoretische Fysica, KU Leuven
\endaffil
\address
Giambattista Giacomin
\hfill\newline
Department of Mathematics
\hfill\newline
Hill Center, Rutgers University
\hfill\newline
New Brunswick, N.J. 08903, U.S.A.
\endaddress
\email
giacomin\@boltzmann.rutgers.edu
\endemail
\address
Joel L. Lebowitz
\hfill\newline
Departments of Mathematics and Physics
\hfill\newline
Hill Center, Rutgers University
\hfill\newline
New Brunswick, N.J. 08903, U.S.A.
\endaddress
\email
lebowitz\@math.rutgers.edu
\endemail
\address
Christian Maes
\hfill\newline
Instituut voor Theoretische Fysica, KU Leuven
\hfill\newline
Celestijnenlaan 200D
\hfill\newline
B3001 Leuven, Belgium
\endaddress
\email
FGBDA35\@BLEKUL11.bitnet
\endemail
\keywords
percolation, Gibbs measures,
nonuniqueness, antiferromagnets,
hardcore models, WidomRowlinson
continuum model
\endkeywords
\abstract
We extend some relations between percolation and the
dependence of Gibbs states on boundary conditions known for Ising
ferromagnets to other systems and investigate
their general validity:
percolation is defined in terms of the agreement of a configuration
with one of the ground states of the system.
This extension is studied via examples and counterexamples, including
the antiferromagnetic Ising and hard core models on bipartite
lattices, Potts, many layered Ising and continuum WidomRowlinson models.
In particular our results on the hard square lattice model make rigorous
observations made in Hu and Mak (1989) and (1990) on the basis of computer
simulations. Moreover we observe that the (naturally defined)
clusters of the WidomRowlinson model play (for the WRmodel itself)
the same role that the clusters of the FortuinKasteleyn
measure play for the ferromagnetic Potts models.
The phase transition and percolation in this system can be mapped into
the corresponding liquidvapor transition of a one component fluid.
\endabstract
\endtopmatter
\document
\head 1. Introduction
\endhead
\baselineskip=25pt
The low temperature phases of matter can generally be thought of
as small
thermal perturbations of corresponding ground states. This is particularly simple for
the case of a classical lattice system whose configuration is specified by
$\{\s(x)\}$ with $x$
on some regular lattice ${\Cal L}$, and $\s(x)$
a spin variable taking one of a finite number of values at each
site.
Given a local interaction having
a finite number of
periodic ground state configurations (PGSC), we can then take these
PGSC as boundary conditions for
the Gibbs measures in $\Lambda$, i.e.
consider the Gibbs distribution at inverse temperature $\beta$
with boundary conditions given
by a PGSC on $\Lambda^c$, the complement of $\Lambda$, of some
box $\Lambda$ (see Section 2 for precise definitions).
For fixed $\Lambda$ and $\beta > 0$ the
boundary condition has a definite influence on the probability
distribution of
the spins in the bulk of $\Lambda$. In particular, in the
limit
$\beta \nearrow \infty$, the Gibbs measure in $\Lambda$
becomes concentrated on the
extension inside $\Lambda$ of
the PGSC imposed outside $\Lambda$. We are interested in knowing
whether this influence persists for $\beta$ large but finite
when we take the volume $\Lambda$ to be macroscopic.
In other words, is there an ordered state in the infinite volume
($\Lambda\nearrow \Cal L$) system
dependent upon the PGSC imposed as boundary conditions on the
finite box $\Lambda$?
\newline
If there is such a {\it memory} of the state with respect to boundary
conditions
{\it at infinity}, one sometimes calls the corresponding ground state
thermally stable. The well known Peierls argument provides
such a result for the ferromagnetic Ising model and the
PirogovSinai theory, Pirogov and Sinai (1976), and its extensions
study the genericity of this scenario. Under certain conditions, e.g.
when there are a finite number of thermally stable periodic ground states, it allows
one to construct the low temperature phase diagram of the system
(for a review see Sinai (1982),
(1986) Zahradn\'{\i}k (1984),
Bricmont and Slawny (1985), Slawny (1986)).
In this note, we investigate the geometric or percolation picture
of this memory effect, considered by many authors (Russo (1979),
Fortuin and Kasteleyn (1972), Georgii (1988)):
to make precise the
intuition that the influence of the boundary conditions must propagate
via ``interacting sites'' from infinity to the center of the
system if the corresponding nonzero temperature state is to be
stable.
We are thus led to the question:
in what
sense is the finite temperature
state corresponding to a ground state accompanied by the presence of
an infinite connected cluster on which the PGSC is
realized.
This question is fully answered at sufficiently low temperatures where
the proof of the existence of different phases, determined by different
GSC boundary conditions, via the Peierls argument or PirogovSinai
theory
actually proves the existence in each such phase of just such a
cluster and no other. Our interest here is therefore primarily
the extension of well known low temperature results to higher
temperatures where
the Peierls and PirogovSinai
arguments fail.
We are particularly interested in the
question of whether, for two dimensional systems with particular
symmetries,
the existence of such an ordered state
is equivalent to the percolation of that and only that
ground state
configuration.
This is known for example in the case of the standard
ferromagnetic Ising model without external magnetic field. As we show
below, this picture extends to other models such as the antiferromagnetic
Ising model, the hard square model and the WidomRowlinson
model. A weaker statement is proven for the Potts model.
However, we will also give examples (see
Section 3(c) below) in which the existence of an ordered state
does not imply percolation of the corresponding ground state.
We emphasize that our setup is different from that
in the FortuinKasteleyn representation.
Here the percolation clusters are defined directly in terms
of the Gibbs state configuration.
Therefore in general we do not expect to have
direct relations between e.g. correlation functions in
the Gibbs state and corresponding percolation
probabilities. Still we will see that for the model
in Section 4 such a relation can in fact be established.
In the next Section we present the general framework.
This is implemented by the examples of Section
3 in the case of lattice systems. Section 4 is devoted
to the continuum WidomRowlinson model.
\head
2. General framework
\endhead
We present the notation here in case of lattice
systems. The continuum model
is contained in Section 4.
\definition{The lattice}
We restrict our attention to the
$d$dimensional
lattice $\Bbb Z^d$, $d\ge 2$. This restriction is made
for notational convenience and possible generalizations
will be noted later on.
By $x\sim y$ we mean that $x$ and $y \in \Bbb Z^d$
are nearest neighbors.
Given $\Lambda \subset \Bbb Z^d$, $\partial \Lambda$
will denote the outer boundary of $\Lambda $,
i.e. $\partial \Lambda
=\{x\in \Bbb Z ^d \setminus \Lambda :
y \sim x \text{ for some } y\in \Lambda\}$.
In the sequel $\Lambda$ is always a finite subset of
$\Bbb Z ^d$.
\enddefinition
\definition{The configuration space}
The single site state space is denoted by $S$ and it contains
a finite number of elements
($\vert
S \vert =q$). An infinite volume configuration $\s = \{\s(x)\}_{ x\in \Bbb
Z^d }
$ is an element of $\Omega=S^{\Bbb Z ^d}$.
A configuration $\sigma $ is periodic if there is a $(k_1,\ldots ,
k_d) \in \Bbb Z ^d$ such that if $y=x+(n_1 k_1 ,\ldots ,n_d k_d)$
for some $(n_1, \ldots , n_d )\in \Bbb Z^d$, then $\sigma (x)=\sigma (y)$.
\enddefinition
\definition{The Hamiltonian and its ground states}
The translation invariant Hamiltonian $H$ consists (for
definiteness) of the nearest neighbor interaction $U:S\times S
\rightarrow {\Bbb R}$
and the selfenergy $V: S \rightarrow{\Bbb R}$
$$
H(\s)=H(\s ;J,h) = J \sum_{x\sim y } U (\s(x),\s(y))  h \sum_x V
(\s(x))
\tag2.1
$$
in which the infinite sums are formal. The real parameters $J\ge 0$
and $h$ in (2.1) play the roles of a coupling coefficient
and magnetic field (or chemical potential).
Consider $\s ,\e \in \Omega$ such that $\sigma (x) =\eta (x)$
for all $x \in {\Bbb Z ^d}\setminus \Lambda$, for some finite
$\Lambda\subset \Bbb Z ^d$.
Then $\s$ and $\e$ are said to be {\it equal at
infinity} or one is an {\it excitation} of the other. Their finite
relative Hamiltonian $H(\s\e)$ is then
$$
\aligned
H(\s\e) &\equiv H(\s)  H(\e)
\\
&\equiv
J \sum_{x\sim y } [U (\s(x),\s(y)) 
U (\e(x),\e(y)) ]
 h \sum_x [V(\s(x))V(\e(x))]
\endaligned
\tag 2.2
$$
in which the sum is now only over a finite number of terms.
A configuration $\e\in \Omega$ is a {\it ground state} if $H(\s\e)\geq0$
for any excitation $\s$ of $\e$.
The set of periodic ground state configurations (PGSC)
is denoted by
$g(H)$.
\enddefinition
\definition{The Gibbs measure}
Take $\e \in g(H)$. The finite volume ($\Lambda$)
Gibbs state with $\e$ boundary conditions is the probability measure on
the product space
$S^\Lambda$
defined by
$$
\mu_{\beta ,\Lambda}^{\e }(\s_\Lambda) =
\frac1{Z_\Lambda (\beta,\e)}
\exp[\beta H(\s_\Lambda^\e\e)]
\tag2.3
$$
for $\sigma _\Lambda\in S^\Lambda$. In (2.3) $\beta >0$ plays
the role of the inverse temperature,
$Z_\Lambda (\beta,\e)$ is the normalization constant and
$$
\s_\Lambda^\e (x) =
\cases
\s _\Lambda (x)& \text{ if } x\in \Lambda\\
\e(x)& \text{ if } x\in \Lambda^c=\Bbb Z ^d \setminus \Lambda
\endcases
\tag2.4
$$
Observe that $\sigma _\Lambda^\e$ is an excitation of $\e$.
The infinite volume
Gibbs measure $\mu _\beta ^\eta$ is
obtained taking the limit as $\Lambda
\nearrow {\Bbb Z ^d}$ along suitable subsequences:
$\mu ^\e _\beta$ is a measure on $\Omega$ endowed with
the product sigma algebra
(for details see Georgii (1988), Simon (1993)).
By phase coexistence at $\beta$ we mean that
$\mu ^\eta _\beta \not= \mu ^\sigma _\beta $
for some $\eta, \sigma \in g (H)$.
\enddefinition
\definition{Agreement with the PGSC}
To understand the geometrical structure of such phases
fix $\eta \in g(H)$ and the map
$s_{\e }:\Omega \rightarrow \{0,1\}^{\Bbb Z ^d}$ expressing
agreement or disagreement with the ground state $\eta$
$$
(s _{\eta } (\sigma))(x)
=
\cases
1 &\text{ if } \s (x) = \e(x) \\
0 &\text{ if } \s(x) \neq \e(x) \\
\endcases
\tag2.5
$$
We denote by
$\nu _{\beta, \Lambda} ^\eta =\mu ^\e _{\beta,
\Lambda} s^{1}_{\eta}$ the measure
on $\{0,1\}^\Lambda$ induced by the map
$s_{\e }$ (with a slight abuse of notation). As before we can take
the limit $\Lambda \nearrow \Bbb Z ^d$; write it as $\nu _\beta ^\eta$.
\enddefinition
\definition{Percolation of the ground states}
Percolation is defined in terms of regular site percolation for
the measure $\nu ^\e _\beta$: for $s \in
\{0,1\}^{\Bbb Z ^d}$ define the clusters
$$
C (s ,\alpha)=\{x \in {\Bbb Z ^d}: s (x)=\alpha \}
\tag2.6
$$
in which $\alpha=0,1$. Call $\overline{C}(\s ,\alpha)$ the maximal
connected component containing the origin of ${\Bbb Z ^d}$
($\overline{C}(s ,\alpha)=\emptyset$ if $s (0)\neq \alpha$).
The probability of $\eta$percolation (respectively non
$\eta$percolation) in the phase with boundary condition $\eta$
is defined as
$$
\theta ^\eta _\beta (\alpha)= \nu^\e _\beta
(\{s: \vert \overline{C}(s ,\alpha)\vert =\infty\})
\tag2.7
$$
with $\alpha =1$ (respectively $\alpha =0$).
We have $\eta$percolation, or agreement percolation,
(respectively non $\eta$percolation)
if $\theta ^\eta _\beta (1) >0$ (respectively $\theta ^\eta _\beta (0)>0$).
Observe that not having $\eta$percolation in general does not imply
having non $\eta$percolation.
\enddefinition
\definition{Attractive measures}
If $S$ is ordered ($\ge$),
$\Omega$
is ordered ($\succ$) by defining
$\sigma ^\prime \succ \sigma $
if $\sigma ^\prime (x) \ge \sigma(x)$ for all
$x \in \Bbb Z ^d$. The order chosen for $S$ may depend
on the site $x$.
A measurable event $A$ is said to be increasing if
$\sigma \in A$ and $\sigma ^\prime \succ \sigma $
implies $\sigma ^\prime\in A$.
A measure $\mu$ on $\Omega$ is attractive (FKG)
if $\mu (A \cap B )\ge \mu (A) \mu (B)$
for all $A$, $B$ increasing events, see Fortuin {\it et al.} (1971).
\enddefinition
We are interested in the verification of three main statements:
\newline
$a)$
we are asking whether phase coexistence
implies percolation, i.e. given $\eta , \eta ^\prime \in g(H)$
is it true that
$$
\mu_\beta ^{\eta} \not= \mu_\beta ^{\eta^\prime }
\Longrightarrow \ \
\theta ^\eta _\beta (1)>0
\tag2.8a
$$
\newline
$b)$ whether the reverse implication is true
when $d=2$
$$
\theta ^\eta _\beta (1)>0 \ \ \Longleftrightarrow \ \
\mu_\beta ^{\eta } \not=\mu_\beta ^{\eta^\prime }
\tag2.8b
$$
\newline
$c)$ still for $d=2$, can we rule
out the possibility of disagreement percolation
$$
\theta _\beta ^\eta (0)=0
\tag2.8c
$$
For the models we will be dealing with, if $d>2$ the statement $(2.8b)$
fails or it is expected to fail (see Aizenman {\it et al.}
(1987) and references therein).
\head 3. Lattice models
\endhead
As already mentioned in the introduction, the case in which
$S=\{1,+1\}$, $U(a,b)=ab$ and $h=0$
(ferromagnetic Ising model)
is well understood.
In this case
$g(H)=\{\e_+ ,\e_\}$ ($\e_+(x)=+1, x\in \Bbb Z ^d$ and
$\e_=\e _+$) and for $\beta >\beta _c$ ($1/\beta _c$ is the
critical temperature)
there are two distinct translation invariant extremal states
obtained by taking $\eta _+$ and $\eta _$
as boundary conditions and the proof
of the statements $(2.8a)$, $(2.8b)$ and $(2.8c)$
can be found in
Coniglio {\it et al} (1976).
\subhead a) Ising antiferromagnet
\endsubhead
$\Bbb Z ^d$ is bipartite, that is
it can be split into two sublattices (in this case
the points with even sum of coordinates $\Bbb Z ^d_e$
and the ones with odd sum $\Bbb Z ^d _o$) such that
if $x,y\in\Bbb Z ^d_e$ (or $x,y\in\Bbb Z ^d _o$) then
$x \not\sim y$.
Fix $S=\{1,+1\}$ and take
$$
U(a,b)=ab \ \ \ \ \ \ \ \ \
V(a)=a
\tag3.1
$$
If $h<2dJ$, then $g(H)=\{ \eta _e , \eta _o\}$
in which $\eta _e (x)=+1$ if $x\in \Bbb Z ^d_e$, $\eta _e (x)=0$
otherwise and $\eta _o =\eta _e$ (see for example Dobrushin and
Shlosman (1985)). The phase transition in this model
has been studied in Dobrushin (1968) and Heilmann (1974).
Because of the bipartite structure,
flipping the spins on the even (or odd) sites makes the
model into a ferromagnetic Ising model with a staggered
magnetic field. In particular
for magnetic field $h=0$, there is the
usual Curie point $T_c$ (the critical temperature for
the ferromagnetic model above).
\proclaim{Proposition 3.1:}
With the choices of (3.1) in (2.1), for any $h$ and any $J \ge 0$,
statements (2.8a), (2.8b) and (2.8c) hold.
\endproclaim
\demo{Proof}
Take $\e =\e_e$. For any $\Lambda $
containing the origin, we have that
if $\mu _\beta ^{\eta _e}\not=\mu _\beta ^{\eta _o}$
$$
\nu_\beta ^{\eta _e}
(s(0)=1 \vert s(x)=0 \text{ for all } x\in {\partial \Lambda} )=
$$
$$
\mu _\beta ^{\eta _e} (\sigma (0)=+1 \vert
\sigma (x)=\eta _o(x),
\text{ for all } x \in \partial \Lambda)=
$$
$$
\mu _\beta ^{\eta _o}
(\sigma (0)=+1 \vert
\sigma (x)=\eta_o(x),
\text{ for all } x \in \partial \Lambda)
\le
\tag3.2
$$
$$
\mu_\beta ^{\eta _o}
(\sigma (0)=+1) < \nu ^{\eta _e} _\beta
(s (0)=+1)
$$
The first equality is a change of notation and the
second one is the Markov property.
The first inequality follows from the attractivity
of the measure $\mu_\beta ^{\eta _o}$
with respect to the order relation
$\sigma \succ \sigma ^\prime$ iff
$\sigma (x)\eta _e(x) \ge \sigma ^\prime (x) \eta _e (x)$
for all $x \in \Bbb Z ^d$.
By Theorem 2 of Bricmont {\it et al.} (1987) we
have $(2.8a)$. \newline
If $\mu _\beta ^{\eta _e}=\mu _\beta ^{\eta _o}$ then,
by the fact that $\mu _\beta ^{\eta _e} T_i^{1}=\mu _\beta ^{\eta
_o}$ ($T_i \sigma (x)=\sigma (xe_i)$, $e_i$ is the unit
vector in the direction $i\in \{1 ,\ldots ,d\}$) we have that
$\nu _\beta ^{\eta _e}=\nu _\beta ^{\eta _o}\equiv \nu _\beta$. This
implies
that $\nu _\beta $, besides being attractive (in the usual order),
is reflection
invariant and ergodic under translations. Hence if $d=2$ the main
Theorem of Gandolfi {\it et al.} (1988) applies and $(2.8b)$ is proven.
If $\mu _\beta ^{\eta _e}\not=\mu _\beta ^{\eta _o}$,
$\mu _\beta ^{\eta }$'s are not $T_i$invariant, but
only $T_i^2$invariant. Hence we have to use an extension (Klein and
Yang (1993)) of the result in Gandolfi {\it et al.} (1988)
to get $(2.8c)$.
\qed
\enddemo
The results of Proposition 3.1 can be extended
to more general bipartite lattices.
\subhead b) Hard core lattices
\endsubhead
The hard core (or hard squares) lattice model with activity
$\lambda$ is defined by the infinite volume limit
($\Lambda \nearrow \Bbb Z^d$) of the measure
$\mu _{\lambda , \Lambda}$ on the product space $\{0,1\}^\Lambda $
$$
\mu_{\lambda , \Lambda}^\gamma (\sigma_\Lambda)=
{\chi (\sigma_\Lambda^{\eta_\gamma}) \lambda ^{N_\Lambda}
\over Z_\gamma (\lambda , \Lambda)}
\tag3.3
$$
In (3.3) $\gamma \in \{e,o\}$ and
$\eta _e(x)=1$ if $x$ is even,
$\eta _e(x)=0$ if $x $ is odd ($\eta _o(x) =1 \eta _e(x)$
for all $x$). Further, $N_\Lambda=
\sum_{x\in
\Lambda}
\sigma_\Lambda^{\eta_\gamma} (x)$
with $\sigma _\Lambda ^\eta$ as defined in
(2.4) ; $Z_\gamma (\lambda ,\Lambda)$ is a normalization and for $\sigma
\in \{0 ,1 \} ^{\Bbb Z ^d}$
$$
\chi(\sigma)=
\cases
1 & \text{ if } \sigma (x) \sigma (y)=0 \text{ for all } x\sim y \\
0 & \text{ otherwise} \\
\endcases
\tag3.4
$$
Hence the model can be seen as a gas of hard (i.e. non overlapping)
squares (see Fig.1) or diamonds with fugacity $\lambda$. Call $C(\eta)=\cup _{x:\eta (x)=1} Q(x)$
($Q(x)=\{y \in \Bbb R ^d : \sum _{i=1} ^d \vert y_i x _i\vert =1\}$)
and denote by $C_0 (\eta )$ the connected component of $C(\eta )$
that contains the origin (two squares touching at a corner are
connected, see Fig.1). There are clearly two different type
of connected components of $C$ : the ones for which the squares are
centered on even sites and the ones for which they
are centered on odd sites. We will call them type $e$ and
type $o$ clusters.
For $\gamma , \delta \in \{ e ,o\}$ define
$$
\theta_\lambda ^{\gamma} (\delta )=
\mu ^\gamma _\lambda (\{ \eta :C_0(\eta) \text{ is of type } \delta
\text{
and unbounded}\})
\tag3.5
$$
(3.5) is clearly the analogue of (2.7).
The hard core model can be seen as a limit of the antiferromagnetic
Ising model for
$\beta \rightarrow \infty$ and $h\rightarrow 2dJ$ along
$\beta (h2dJ)=\text{cotan} \theta$ ($\theta \in (0, \pi)$).
The phase diagram point $(2dJ,0)$ is highly degenerate, since
there are infinitely many (in general nonperiodic)
ground states.
In these limits map $\sigma \in \{1, +1 \}^{\Bbb Z ^d}
\rightarrow \eta \in \{ 0,1 \}^{\Bbb Z ^d}$ by setting
$\eta (x)=1$ if $\sigma (x)=1$ and $\eta (x)=0$ if
$\sigma (x)=+1$. We get a hard square model with activity
$\lambda = \exp (2 \text{cotan} \theta )$ (see Dobrushin {\it et al.}
(1985)
for details). This picture suggests that the critical
fugacity (if it exists) should correspond to $\theta _c$
(see Fig.2).
We rephrase $(2.8a)$, $(2.8b)$, $(2.8c)$ into
\proclaim{Proposition 3.2} For the hard square model
$$
\mu _\lambda ^e \not= \mu _\lambda ^o
\Longrightarrow
\theta^e _\lambda (e) >0
\tag3.6a
$$
Moreover
if $d=2$
$$
\theta^e _\lambda (e) >0 \Longleftrightarrow
\mu _\lambda ^e \not= \mu _\lambda ^o
\tag3.6b
$$
and
$$
\theta_\lambda ^e(o)=0
\tag3.6c
$$
\endproclaim
\demo{Proof}
Define the partial order
$\eta ^\prime \succ \eta$ if
$\eta ^\prime (x) \ge \eta(x)$ for
$x\in \Bbb Z^d _e$
and $\eta ^\prime (x) \le \eta(x)$
if $x \in \Bbb Z^d _o$. It is easy to check
that $\mu _{\lambda, \Lambda}$ and its infinite volume limit
are attractive. The proof is then the same of the
one of Proposition 3.1.
\qed
\enddemo
\remark{Remark}
The motivation for Proposition 3.2 was provided by
the work of C.K. Hu and K.S. Mak (Hu and Mak (1989) and (1990)) in which
a similar result is conjectured on the basis of
computer simulations. In Hu and Mak (1989) and (1990)
they discuss also the case of hard core particles
on a triangular lattice, the hard hexagon model.
Our result extends easily
to other bipartite lattices, such as the hexagonal one.
The triangular lattice with nearest neighbor bonds
is not bipartite, so our proof
does not work. In view of Hu and Mak (1989) and (1990), one expects
that Proposition 3.2 still holds for this model,
but it is unclear to us, especially in view of the results of
the Section 3.d on {\sl many layer} Ising models, whether Proposition 3.1
($d=2$) holds for the whole domain of coexisting phases of the
antiferromagnetic Ising model on the triangular lattice (in this case,
for $h \in (0,6J)$, $g(H)$ contains three configuration).
\endremark
\subhead c) {\sl Many layer} models
\endsubhead
Given a model with configuration space $\{1, +1\}^{\Bbb Z ^d}$
and Hamiltonian $H_1$ we can define a family of new models indexed
by integers $Q\ge 2$.
Take $S=\{1 ,+1\}^Q$
so that the infinite volume configuration
is of the form
$\xi =(\sigma _1,\ldots ,\sigma _Q)\in S^{\Bbb Z^d}$
(to be identified with $\{1, +1\}^{\Bbb Z^d}
\times \ldots \times \{1, +1\}^{\Bbb Z^d}$ ($Q$ copies)).
Define the formal Hamiltonian as
$$
H(\xi)=\sum _{i=1}^Q H_1(\sigma _i)
\tag3.7
$$
so that for boundary conditions $\omega=
(\eta _1 , \ldots ,\eta _Q)$, $\mu ^\omega _{\beta, \Lambda}=
\mu^{\eta _1}_{1,\beta ,\Lambda} \times \ldots \times
\mu^{\eta _Q}_{1,\beta ,\Lambda}$ (where the subscript 1 refers
to the system with Hamiltonian $H_1$) is the finite volume
Gibbs state with respect to $H$.
Observe that $g(H)=(g(H_1))^Q$ (with the previous identification).
\subsubhead Duplicated Ising model \endsubsubhead
Take $Q=2$ and $H_1$ as in (2.1), characterized by
$U(a,b)=ab$ and $h=0$.
As observed before $g(H_1)=\{ \eta _+ ,\eta _\}$ and
so $g(H)=\{ \omega _{++}, \omega _{+}, \omega _{+}, \omega _{}\}$,
where $\omega _{++}= (\eta _+ ,\eta _+)$ and so on.
\proclaim{Proposition 3.3}
For the duplicated Ising model statements (2.8a), (2.8b)
and (2.8c) hold.
\endproclaim
\demo{Proof}
We want to estimate the expectation value of the sum of the spins
at the origin given that
$\xi (y)=(\sigma _1(y), \sigma _2(y))\not=
(+1, +1)$ for $y \in \partial \Lambda$.
In that case
$\sigma _1 (y)+ \sigma _2(y) \le 0$ for $y\in \partial \Lambda$,
so that by attractivity
$$
\mu_\beta ^{\omega _{++}}
(\sigma _1(0)+\sigma _2(0) \vert
s_{\omega _{++}} (\xi)=0
\text{ on } \partial \Lambda ) \le
$$
$$
\mu_\beta ^{\omega_{++}}
(\sigma _1(0)+\sigma _2(0)\vert
(\sigma _1, \sigma _2): \sigma _2(y)=\sigma _1(y)
\text{ for all } y \in \partial \Lambda)
=
\tag3.8
$$
$$
\sum_{\sigma\in \{1,+1\}^{\partial \Lambda }} \Big\{
\left[\mu _{1, \beta}
^{\sigma} (\sigma _1(0))+ \mu _{1,\beta} ^{\sigma} (\sigma _2(0))\right]
\cdot
$$
$$
\mu_\beta ^{\omega _{++}}
(\xi(x)=(\sigma_1(x),\sigma_1(x)), x\in\partial\Lambda \vert
s_{\omega _{++}} (\xi) = 0
\text{ on } \partial \Lambda )\Big\}
= 0
$$
On the other hand if $\mu_{1, \beta }^{\eta _+} \not= \mu _{1,
\beta}^{\eta _}$
$$
\mu ^{\omega _{++}}_\beta (\sigma _1(0)+\sigma _2 (0))
\equiv 2m^{\star }(\beta)>0
\tag3.9
$$
Apply Theorem 2 of Bricmont {\it et al.} (1987)
to get $(2.8a)$. For what concerns $(2.8b)$ and $(2.8c)$,
it is straightforward to see that $\nu ^{\omega _{++}}_\beta=
\mu ^{\omega _{++}}_\beta s_{\omega
_{++}} ^{1}$
is reflection invariant, ergodic under translation and attractive.
Hence we can apply Gandolfi {\it et al.} (1988)
to get $(2.8b)$ and $(2.8c)$.
\qed
\enddemo
\remark{Remark}
Note that as the phase transition is second order ($\mu^{\eta
_+}_{1,\beta}(\s(0))^+ \equiv
m^{\star}(\beta)$ is continuous at $\beta=\beta_c$), the density of $(+1,+1)$
just below the critical
temperature is only slightly above 1/4 and still the
$(+,+)$ spins percolate in the $(+,+)$ state\footnote{The threshold for
Bernoulli site percolation on $\Bbb Z^2$ is about 0.59}. In the same way,
the density of sites $x\in\Bbb Z^2$ where $(\s_1(x),\s_2(x))\neq(+,+)$ is
there only slightly below 3/4 and still, in the $(+,+)$ state, they do not
percolate.
\endremark
The question therefore arises whether one can go arbitrarily far and
construct examples where there is percolation for arbitrarily low densities
or where there is no percolation no matter how large one makes the density.
Such examples in fact exist (see for example Molchanov and Stepanov (1983))
but they are rather singular. It may well be that a minimum
density for having percolation actually exists for {\it good} Markov
fields.
\proclaim{Conjecture}
Given $\mu$ on the product space $S^{\Bbb Z^d}$ ($\vert S \vert =q
<\infty$), a translation invariant pure Gibbs state for some isotropic nearest
neighbor interaction, there is a constant $c(d)>0$ (independent
of $q$) such that, if $\mu (\sigma (0)=a) \beta_c$ for which there is phase
coexistence.
There is $Q(\beta,d)$ such that for all $Q\ge Q(\beta,d)$
$$\theta ^{\omega}_\beta (1)=0
\tag3.10
$$
\endproclaim
\demo{Proof}
By direct computation
$$
\max_{\xi^\prime}
\mu^{\omega}_\beta ( \xi (x)=(1,...1)\vert
\xi^\prime (y), y\sim x)=
\left({1 \over 1+\exp(4d\beta J)}\right)^Q
\tag3.11
$$
we can now take $Q$
sufficiently large so that
$$
\left(
{1 \over 1+\exp (4d\beta J)}\right)^Q
1/2$ and $\beta_c=
(1/2J)\log(1+(\sqrt{5}/2))$ we get $Q(\beta_c,2)=25$).
\qed
\enddemo
Hence we have an example in which the measure is attractive,
but nevertheless phase coexistence does not imply
percolation. Of course Proposition 3.4 also holds
for the {\sl many layer} version of other Markov fields.
\noindent
\remark{Remark}
If in $\mu^{\omega _{+}}_\beta$ there is with probability
one some
circuit $\partial \Lambda $ around the origin on which the two coordinates
agree, i.e. $\s_1(y)=\s_2(y), y\in \partial \Lambda $, then
$\mu^{\eta _+}_{1,\beta}=\mu^{\eta_}_{1,\beta}$ (the effect of the
boundary will be clearly cancelled by conditioning on the circuit $\partial
\Lambda$).
In other words if $\mu^{\eta _+}_\beta\neq\mu^{\eta_}_\beta$ (respectively
$\mu ^{\eta _1}_{1,\beta} \neq \mu ^{\eta_2}_{1,\beta}$ for $\eta _1, \eta
_2 \in \Omega$), there must be
{\it disagreement percolation}
in $\mu^{\omega _{++}}_\beta$ (respectively in $\mu^{\eta_1}_{1, \beta}
\times \mu ^{\eta _2}_{1, \beta}$), as it is done in
van den Berg (1993) (see also van den Berg and Maes (1994) for another
coupling). {\it Disagreement percolation}
means that there is an infinite cluster on which $\s_1(x)\neq\s_2(x)$.
This is
applied in
van den Berg and Steif (1994) for the hard core model of above.
They take independently two realizations $(\s_1,\s_2)$ according to the
product coupling $\mu^e_\lambda\times\mu^o_\lambda$.
A site $x \in \Bbb Z^2$ is a site of disagreement if
$\s_1(x)\neq\s_2(x)$. They prove that $\mu^e_\lambda=\mu^o_\lambda$ if and
only if
$\mu^e_\lambda\times \mu^o_\lambda(\{(\s_1,\s_2)$ has an infinite path of
disagreement$\})=0$.
Using our general formulation we can strengthen this result somewhat : if
$\mu^e_\lambda \neq\mu^o_\lambda $ not only will we get disagreement
percolation in
the above sense but in the state $\mu^e_\lambda \times\mu^o_\lambda $ this
percolation will be over sites $x$ where
$$\aligned
(\s_1(x),\s_2(x)) & = (\eta_e(x),\eta_o(x))\\
& = (1,0) \text{ for } x
\text{ even }\\
& = (0,1) \text{ for } x \text{ odd }
\endaligned
\tag3.13$$
The reason is that $\mu^e_\lambda \neq\mu^o_\lambda $ is equivalent to
$\mu^e\times\mu^o\neq\mu^o\times\mu^e$ implying
the stability in $\mu^e_\lambda \times\mu^o_\lambda $ of the configuration
$(\eta_e,\eta_o)$ as given in (3.13).
\endremark
\subhead d) The qstate Potts model\endsubhead
In this case $S=\{1, \ldots ,q\}$.
The Hamiltonian (2.1)
is specified by taking $J > 0, h=0$ and $U(a,b) =
0$ if $a=b$,
$U(a,b)=1$ if $a\neq b$.
\newline
It is straightforward to see that $g(H)=\{\eta _a :a \in S\}$
where $\eta _a(x)=a $ for all $x \in {\Bbb Z }^d$.
A very useful way to analyze the Potts model is to take the FKrepresentation
of Fortuin and Kasteleyn (1972). For that we let $\ell(b) = 0, 1$ be a
bond configuration. A bond $b = $ is connecting nearest
neighbors $x\sim y\in \Bbb Z^d$
and it can be {\it open} ($\ell(b)=1$) or {\it closed} ($\ell(b) = 0$).
In $\Lambda\subset\Bbb Z^d$ we fix a bond configuration $\ell$
by assigning to all bonds $b=$ (connecting nearest neighbors $x\sim y$ at
least one of which is inside $\Lambda$) the value 1 or 0.
For bond percolation see the definitions in Grimmett (1994).
We define the following expectation for local
functions $f(\s_\Lambda)$ of the Potts model variables $\s_\Lambda$ in the
volume $\Lambda$ with boundary conditions $\xi$ :
$$
\langle f \rangle_{\Lambda}^{\xi}(\ell) = \frac1{q^{n_{\Lambda}(\ell)}}
\sum_{\s_{\Lambda}}
f(\s_{\Lambda})
\prod_{b=\cap\Lambda\neq\emptyset : \ell_b = 1} \delta(\s_\Lambda^\xi(x),
\s_\Lambda^\xi(y))\tag3.14$$
%
Here $\delta$ is the Kronecker delta, $n_{\Lambda}(\ell)$ is the number of
connected $\ell$clusters in the
volume
$\Lambda$ so that the expectation (3.7) is normalized.
The configuration $\s_{\Lambda}^\xi$ is defined as in (2.4). The reason for
introducing (3.7) is that the Potts model expectations in volume $\Lambda$
with boundary conditions $\xi$ can be written as
$$
\mu_{\beta, \Lambda}^\xi (f)=\frac1{Z_\Lambda(\beta,\xi)} \sum_{\ell}
\prod_{b\cap\Lambda\neq\emptyset} p^{\ell_b} (1p)^{1\ell_b}
q^{n_{\Lambda}(\ell)} _{\Lambda}^\xi(\ell)\equiv
\nu^{FK}_{q,\beta ,\Lambda} (_{\Lambda}^\xi(\cdot))
\tag3.15$$
%
when we put $p=1e^{\beta J}$. Here $\nu^{FK}_{q,\beta ,\Lambda}$
denotes the finite volume FortuinKasteleyn measure (or random
cluster measure) on the bond configurations whose weights are
defined by (3.15). The infinite volume measure will be denoted by
$\nu ^{FK}_{q,\beta}$ (see Grimmett (1994) and (1994')).
\proclaim{Proposition 3.5}
For the Potts model (2.8a) holds. Moreover
if $d=2$ and $\mu^\eta_\beta \neq \mu ^{\eta ^\prime}_\beta$
for $\eta , \eta ^\prime \in g(H)$ then
$$
\theta_\beta ^\eta (0)=0
\tag3.16
$$
\endproclaim
\demo{Proof}
(2.8a) is easily proven by observing that
$$\multline
\mu ^{\eta _a}_{\beta, \Lambda} (\sigma (0)=a)=\\
{1\over q}+\left({q1 \over q}\right)
\nu ^{FK}_{q, \beta ,\Lambda}(\{\ell:
0 \text{ is connected to }\partial \Lambda \text { by a chain
of open bonds}\})
\endmultline
\tag 3.17
$$
and that given the coupling between $\sigma$ and $\ell$ implicit in
(3.14) and (3.15), $\sigma (x)=1$ if $x$ belongs to one of the bonds
in the infinite cluster of open bonds. By (3.17) the latter exists
a.s. in
the coexistence region (see FortuinKasteleyn (1972) and Aizenman
{\it et al.} (1988)).\newline
To prove (3.16),
let us consider the FK measure $\nu^{FK}_{q,\beta}$
associated to the extremal Potts measure $\mu_\beta ^{\eta_a}$.
>From its construction (see Grimmett (1994) and (1994')), besides being
translation and rotation invariant, it is ergodic under translations.
Moreover it is known that this measure is also attractive (Fortuin and
Kasteleyn (1972), Aizenmann {\it et al.} (1988), Grimmett (1994) and
(1994')).
A straightforward adaptation of Gandolfi {\it et al.} (1988) allows to
conclude
that if there is percolation of open bonds, outside any box containing
the origin there is a circuit of open bonds surrounding the origin
(a.s.). Again by the coupling between $\sigma $ and $\ell$, we conclude
that if $\theta ^{\eta _a} (1)>0$, then $\mu ^{\eta _a} _{\beta}$a.s.
every point is surrounded by a circuit on
which $\sigma(x)=a$. By $(2.8a)$ we conclude.
\qed
\enddemo
\remark{Remark}
Note that in $d=2$ for $q>4$
the ``magnetization'' $\mu^a[\s(0)=a]$ is believed to be discontinuous
at
$\beta=\beta_c$, see Baxter (1982). Therefore, in the twodimensional
Potts model
the lowest density of a ground state configuration for which we know there
is
percolation
is $1/4 + \epsilon$ (for arbitrary $\epsilon > 0$) and is obtained for
$q=4$ in the corresponding Gibbs
state just below the critical
point\footnote{This is similar to the case of
duplicated Ising variables, see example 3(c)}.
\endremark
\subhead e) WidomRowlinson lattice model\endsubhead
The statements $(2.8a)$, $(2.8b)$ and $(2.8c)$ hold also
for a class of models first introduced in Wheeler and Widom (1970).
They are ``spin 1'' models with single site state
space $S=\{1,0,+1\}$ and Hamiltonian (2.1) determined by $U(a,b)=ab(1ab),
V(a)=a^2, 0\leq J <\infty$ and $h > 0$. This model
was shown to have a phase transition by Lebowitz
and Gallavotti (1971) and to be attractive by Lebowitz and Monroe
(1972).\newline
The detailed analysis of these models follows along
standard lines.
\head 4. Continuum model
\endhead
The continuum WidomRowlinson (WR) model (Widom and Rowlinson (1970)
consists of particles of type
A and type B having positions in $\Bbb R ^d$ and fugacities $z_A$ and
$z_B$ whose interaction consists of the hard core constraint
that the centers of any two particles
of different type must be at least distance $R$ from each other.
In Cassandro {\it et al.} (1973) it is shown how this model can be obtained
from a lattice model of the type described in Section 3(e) above.
\newline
More precisely, we take $\Lambda \subset \Bbb R^d$ a finite Borel set
and let $x=(x_1,\ldots ,x_{N_A})$ (respectively
$y=(y_1,\ldots ,y_{N_B})$ denote the position of particles A
(respectively B),
$x_i,y_i \in \Lambda$ for $i=1, \dots ,N_A$ and $j=1, \ldots, N_B$.
Call $X$ the space of
$\sigma$finite integer valued measures over $\Lambda$ (and its
Borel sets);
our probability space will be $\Omega=X\times Y$ ($X=Y$ will have the
topology of weak convergence, that characterizes its Borel sets).
By separability, any element of $X$ can be written as $\sum_{i\in I}
\delta
_{x_i}$ ($I\subset\Bbb Z, \vert I \vert <\infty$) and so we will use the
notation
$N_A(\omega)$, $N_B(\omega)$, $x(\omega)$ and $y(\omega)$ for $\omega
\in\Omega $ with obvious meaning.
The constraint that is imposed is determined by the hard core length $R$:
$$
\min_{i,j}
\vert x_i y_j\vert >R
\tag 4.1
$$
Letting $I[x,y]$
denote the indicator function corresponding to (4.1), we put
$$
Z^\Lambda_{N_A ,N_B}=
\int_{\Lambda ^{N_A} \times \Lambda ^{N_B}}
\roman{d} \lambda _{N_A}(x)
\roman{d} \lambda _{N_B}(y)
I[x,y]
\tag4.2
$$
for
$
\roman{d} \lambda _{N}(x) =
\roman{d} ^d x_1 \ldots \roman{d}^d x_N
$ the $N$product Lebesgue measure.
Fixing the fugacities $z_A , z_B>0$,
the grandcanonical partition function of the WRmodel
is then
$$
\Xi=\Xi(\Lambda ,z_A, z_B)=
\sum _{N_A ,N_B}
{z_A ^{N_A} z_B^{N_B} \over N_A ! N_B!}
Z^\Lambda _{N_A ,N_B}
\tag 4.3
$$
We will be mostly interested in the case $z_A=z_B=z$, for which we adopt
the notation $\Xi (\Lambda, z)$.\newline
So far we have not spoken of boundary conditions. Obviously we can fix the
position of some particles introducing extra constraints
(beyond (4.1)). For example, we speak of boundary conditions
of type A if we replace $I[x,y]$ in (4.1)
by $I_A[x,y]=I[x,y] I_A[y]$ where $I_A[y]$
is the indicator function corresponding to
$$
\inf_{j,x\in \Lambda ^c}
\vert y_j x \vert >R
\tag 4.4
$$
Analogous definitions and notations apply for boundary conditions of type
B.
The grandcanonical partition function is then changed
into $\Xi_\gamma$, corresponding to the boundary conditions
of type $\gamma=A,B$.
\newline
The finite volume Gibbs measure $\mu ^\gamma _\Lambda$ for boundary
conditions $\gamma =A,B$ gives the probability of finding the
particles in certain regions of $\Lambda$.
If we condition on having $N_A$ type $A$ particles and $N_B$ type $B$
particles in $\Lambda$, the random field will have density
$$
{
\roman {d} \mu_{\Lambda}^\gamma (\cdot) \vert_{N_A(\omega)=
N_A, N_B(\omega )=N_B}
\over \roman{d} \lambda_{N_A} \times \roman{d} \lambda_{N_B}
}(\omega)={1 \over
Z^\Lambda _{N_A ,N_B}
}
{z_A^{N_A} z_B^{N_B} \over
N_A ! N_B!} I_\gamma (\omega)
\tag4.5
$$
in which $I_\gamma (\omega)=I_\gamma[x(\omega),y(\omega)]$
($\omega \in \Omega$) and analogously for $I(\omega)$, see (4.1).
The infinite volume measures are denoted by $\mu ^\gamma$
and can be obtained as limit from $\mu_\Lambda ^\gamma$
as $\Lambda \nearrow \Bbb R^d$ along suitable subsequences
(for examples along spheres).
The measure $\mu ^\gamma$ does depend on the boundary condition $\gamma$
if $z_A= z_B=z$ and $z$ is sufficiently large, Ruelle (1971).
\definition{Clusters and percolation probability}
Define the function $Sp:\Lambda \times \{\omega: I(\omega )=1\}
\rightarrow \{A,B,W\}$ as
$$
Sp(x)=
Sp(x, \omega)=
\cases
A & \text{if dist}(\bold x (\omega),\{x\}) < R/2 \\
B & \text{if dist}(\bold y (\omega),\{x\}) < R/2 \\
W & \text{otherwise}
\endcases
\tag4.6
$$
We can imagine the function $Sp$ a coloring of $\Lambda $ in red (A), black
(B)
or white (W). From now on take $\gamma , \delta \in \{A, B\}$. The $\gamma$ cluster
at the origin ($C^0_\gamma (\omega)$)
will then be defined as the connected component of
$Sp(\cdot ,\omega )^{1}(\gamma)$ that contains
the origin ($C^0_\gamma =\emptyset$ if
$Sp(0) \not= \gamma$).
The percolation probability is thus defined as
$$
\theta ^\gamma (\delta)=
\mu^\gamma (
\{ \omega :
\text{diam}(C^0_\delta (\omega ))=\infty\})
\tag4.7
$$
\enddefinition
\proclaim{Proposition 4.1}
Using the notations and definitions above with $z=z_A=z_B$,
$$
\mu ^A (Sp(0)=A)
\mu ^B (Sp(0)=A)=
\theta ^A(A)\theta^A(B)
\tag4.8
$$
implying
$$
\theta ^A(A) >0
\Longleftarrow \mu^A\neq\mu^B
\tag4.9a
$$
In $d=2$,
$$
\theta^A(B)=0\tag4.9b$$
and
$$
\theta ^A(A) >0
\Longleftrightarrow \mu^A\neq\mu^B
\tag4.9c
$$
\endproclaim
\remark{Remark 1} (4.8) says that the particle clusters in the WRmodel
play a similar role as the random clusters in the FKrepresentation of the
Potts model. We believe this to be the first example where such a direct
relation between the `order parameter' and the cluster geometry is found.
Note also that by attractivity (Lebowitz and Monroe (1972)) the left hand
side of
(4.8) is zero if and only if the $A$phase is different from the $B$phase.
\endremark
\remark{Remark 2} We believe that the results in the Proposition partially
extend to the case
where a hard core condition is added between alike particles. In that case
the measures are not attractive but, as will become
clear from the proof, that is not at all crucial here. Since there are
extra technicalities and quite a bit of extra notation related to the
existence of infinite volume limits if we don't have FKG, we
prefer to give the proof only in the case of strict hard cores.
\endremark
\remark{Remark 3} So far we have assumed for simplicity that the fugacities
of the two types of particles are equal, $z_A=z_B=z$. Note however that
if say $z_A \geq z$, then $\mu^A_{z_A,z}$ stochastically dominates
$\mu^A_{z,z}$ where we now explicitly indicate by subscripts the fugacities
of $A$ respectively, $B$type particles. This implies that if in
$\mu^A_{z,z}$
there is percolation of $A$type particles (as in the phase
coexistence regime, $z>z_c$), then we get the same result for all
$A$particle fugacities $z_A \geq z$. Suppose we now integrate out the
positions of the $B$particles. A simple calculation shows that we get a
new measure for the $A$particles
where now $z$ (previously the fugacity of the $B$particles) plays the role
of an inverse temperature. In that measure, for $z>z_c$, the $A$particles
percolate for all values $z_A \geq z$.
\endremark
\remark{Idea of the proof} Consider an $A$cluster covering the
origin. If it is
bounded, then, by the hard core constraint, there is necessarily a white
region surrounding it, see Fig. 3. This effectively screens the origin
from the external boundary condition. Therefore this contribution to the
probability of having an $A$particle at the origin is the same in all
states. What remains is the probability that the Acluster extends
infinitely far. This idea can be most easily implemented through a
discretization of the space.
\endremark
\demo{Proof of Proposition 4.1}
Take $\Lambda ^\prime \subset \Lambda \subset\Bbb R ^d$
two spheres.
Define the {\sl finite volume} percolation probability
as $ \theta ^\gamma (\delta; \Lambda ,\Lambda ^\prime)=
\mu ^{\gamma}_\Lambda (C^0_\delta (\omega ) \cap \partial \Lambda ^\prime
\not=\emptyset )$ and observe
that
$$
\lim _{\Lambda ^\prime \nearrow \Bbb R^2}
\lim _{\Lambda \nearrow \Bbb R^2}
\theta ^\gamma (\delta; \Lambda ,\Lambda ^\prime )
=\theta ^\gamma (\delta)
\tag4.10
$$
Take $\epsilon > 0 $ small and cover $\Bbb R^d$
with a grid of spacing $\epsilon$. Disregarding boundary problems,
this naturally defines a partition
of $\Bbb R^d$ into squares of sidelength $\epsilon$
($\epsilon$squares).\newline
To control the errors made by the space discretization we define
$$
G(\epsilon)=
\{\omega :
\text{dist}(\bold x ,\bold y) >R+6d\epsilon,
\ \vert \text{dist}(\bold x ,\partial \Lambda ^\prime )R \vert>
6d\epsilon,
\ \vert \text{dist}(\bold y ,\partial \Lambda ^\prime )R \vert>
6d\epsilon, $$
$$
\vert \vert x_i x_k \vert R \vert >6d\epsilon \
i\not= k\in \{1, \ldots ,N_A(\omega)\}
\}
\tag4.11
$$
A simple argument by contradiction
yields the existence of $\Delta_\Lambda (\epsilon) \geq 0$, vanishing
as $\epsilon\downarrow 0$ and such that
$$
\mu^{\gamma}_\Lambda (G(\epsilon))>1\Delta_\Lambda (\epsilon)
\tag4.12
$$
with $\gamma \in \{A, B\}$.
The union of a finite number of $\epsilon$squares is
an $\epsilon$cluster at the origin if
the interior of this set is connected and if it contains the origin.
Denote by ${\Cal C}_\epsilon $ the set of $\epsilon$clusters
at the origin ($\emptyset \in{\Cal C}_\epsilon$).
\newline
We are now going to define a subset of $\Omega$ characterized
by having a certain element of ${\Cal C}_\epsilon$
as minimal $\epsilon$covering of the corresponding maximal
cluster at the origin. More precisely, given $C\in {\Cal C}_\epsilon$
define
$$
\Cal A_\epsilon (C)=\{\omega:
C^0_A(\omega ) \subset C \text{ and for all }
C^\prime \in
{\Cal C}_\epsilon,
C^\prime \subsetneq C,
C^0_A(\omega ) \not\subset C^\prime \}
\tag4.13
$$
We have then by construction that the probability to find an $A$particle
at the origin is
$$
\mu_\Lambda ^A (Sp(0)=A)=
\sum _{C\cap \partial \Lambda ^\prime =\emptyset}
\mu ^A_\Lambda (\Cal A_\epsilon(C))
+
\sum _{C\cap \partial \Lambda ^\prime \not=\emptyset}
\mu ^A_\Lambda (\Cal A_\epsilon(C))
\tag4.14
$$
where the sums are over $C\in {\Cal C}(\epsilon)$.
We deal with the two terms in the right hand side
of (4.14) separately.
First of all observe that by (4.12)
$$
\left\vert
\sum _{C\cap \partial \Lambda ^\prime =\emptyset}
\mu ^A_\Lambda (\Cal A_\epsilon(C))
\sum _{C\cap \partial \Lambda ^\prime =\emptyset}
\mu ^B_\Lambda (\Cal A_\epsilon(C))
\right\vert
$$
$$
\le 2 \Delta_\Lambda (\epsilon)+
\left \vert \sum _C
\left(
\mu ^A_\Lambda (\Cal A_\epsilon(C)\cap G(\epsilon))
\mu ^B_\Lambda (\Cal A_\epsilon(C)\cap G(\epsilon))
\right)
\right\vert
\tag4.15
$$
Define $\partial C_W$ to be that subset of $\Omega$ such that
$Sp(x,\omega)=W$ for all $x$ contained in an
$\epsilon$square adjacent to the outer boundary of $C\in\Cal C_\epsilon
$ (that is the external connected component of the boundary). Since $\Cal
A_\epsilon (C)\cap G(\epsilon) \subset
\Cal A_\epsilon (C)\cap \partial C_W$ we can continue (4.15) obtaining
$$
\left\vert
\sum _{C\cap \partial \Lambda ^\prime =\emptyset}
\mu ^A_\Lambda (\Cal A_\epsilon(C))
\sum _{C\cap \partial \Lambda ^\prime =\emptyset}
\mu ^B_\Lambda (\Cal A_\epsilon(C))\right\vert
$$
$$
\le 4 \Delta_\Lambda (\epsilon)+
\sum_C
\left\vert
\mu ^A_\Lambda (\Cal A_\epsilon(C)\cap \partial C_W )
\mu ^B_\Lambda (\Cal A_\epsilon(C)\cap \partial C_W )
\right\vert
\tag4.16
$$
By the Markov property of the $\mu^\gamma_\Lambda $
the sum in the right hand side of (4.16) is zero.
This is because by conditioning it is straightforward to see that
every term in this sum is equal to
$$
{
\mu_\Lambda
(\Cal A_\epsilon(C) \vert\partial C_W)
\mu_\Lambda
(I_A\vert\partial C_W)
\mu_\Lambda
(\partial C_W)
\over
\mu_\Lambda
(I_A)
}
$$
$$

{
\mu_\Lambda
(\Cal A_\epsilon(C) \vert\partial C_W)
\mu_\Lambda
(I_B\vert\partial C_W)
\mu_\Lambda
(\partial C_W)
\over
\mu_\Lambda
(I_B)
}=0
\tag4.17
$$
The last equality follows because of the symmetry
under exchange $A\leftrightarrow B$.
Hence
$$
\vert
\sum _{C\cap \partial \Lambda ^\prime =\emptyset}
\mu ^A_\Lambda (\Cal A_\epsilon(C))
\sum _{C\cap \partial\Lambda ^\prime )^c=\emptyset}
\mu ^B_\Lambda (\Cal A_\epsilon(C))\vert
\le 4 \Delta_\Lambda (\epsilon)
\tag4.18
$$
The absolute value of the difference between the
second term in the right hand side
of (4.14) and $\theta ^A(A;\Lambda ,\Lambda ^\prime)$ is
by the definition (4.11) of $G(\epsilon)$
smaller than $\Delta _\Lambda (\epsilon)$.\newline
Combining this with
(4.18) and writing (4.14) also for the measure $\mu^B_\Lambda $,
we get
$$
\vert
\mu _\Lambda ^A (Sp(0)=A)
\mu _\Lambda ^B (Sp(0)=A)
+\theta^B(A;\Lambda ,\Lambda ^\prime)\theta^A(A;\Lambda ,\Lambda ^\prime)
\vert \le 6\Delta _\Lambda (\epsilon)
\tag4.19
$$
Hence the left hand side of (4.19) is zero.
The result follows by taking the limits as indicated
in (4.10). In the case of two dimensions ($d=2$), we use
a generalized
version of Gandolfi {\it et al.} (1988) extending their result without
difficulty to the continuum. This
ensures that $\theta ^B(A)=0$ if $\theta ^A(A)>0$.
\qed
\enddemo
\noindent{\bf Acknowledgments} : We are grateful to M. Aizenman, C.K.
Hu, C.M. Newman, L. Russo and A. Sokal for fruitful discussions.
This work is partially supported by NSF grant NSFDMR9213424
and by EC grant CHRXCT930411.
C.M. acknowledges support from the Belgian National Science Foundation
and the hospitality of the Newton Institute in Cambridge. G.G.
acknowledges the support of the C.N.R. and the hospitality of
KU Leuven (Instituut voor Theoretische Fysica).
\noindent
{\bf Note:}
after the completion of this work, we learnt that
J.T. Chayes, L. Chayes and R. Koteck\'y have obtained results
similar to those of the Section 4 of this paper
(J.T. Chayes, L.Chayes, R. Koteck\'y {\it The Analysis of
the WidomRowlinson Model
by Stochastic Geometric Methods}, Preprint (1994)).
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\vfill \eject
Caption for fig.1
\vskip 0.8 cm
\noindent
A portion of a configuration of
the hardsquare lattice gas. The squares
centered on odd sites are painted darker than
the ones centered on even sites. In this figure
there are three odd clusters and two even clusters.
\vskip 2 cm
Caption for fig.2
\vskip 0.8 cm
\noindent
The expected phase diagram of the antiferromagnetic
Ising model for $d=2$. The shadowed area
is the region of phase transition (this is only a qualitative
diagram). The hard square limit is obtained
taking $J \rightarrow \infty$ along the line
with slope $\tan (\theta)$ shown on the figure.
The phase transition for the hardsquare model is expected
to happen at $\theta =\theta _c$.
\vskip 2 cm
Caption for fig.3
\vskip 0.8 cm
\noindent
A portion of a configuration of $A$ (darker) and
$B$ (lighter) particles. There is an $A$cluster at the origin
and it is separated from the other clusters by a white layer.
\vfill\eject
\enddocument